But the LHS is just $1/X+1/Y = 1/S$, so the last inequality is equivalent to the lemma.

Lemma Interpretation

One thing that I was amazed by is that the lemma has such a striking form. If we let $A(x_1,\cdots,x_k)$ be the arithmetic mean of $x_1,\cdots, x_k$, and $H(x_1,\cdots,x_k)$ be its harmonic mean, the lemma can be rewritten as: